Ok, so what does canon tell us about the NX-01's later years.
-Dedication Plaques and the Holodeck of the Enterprise-D tell us that The NX-01 was never in Federation service. So she had to be retired by 2161.
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Strange New Worlds firmly established that there were no Starships named Enterprise between the NX-01 and the 1701.
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These are the Voyages* taking place in 2161, the episode shows us the standard configuration of the NX-01 on various monitors, while also showing the standard configuration during the closing montage, which could be taken out of the context of the episode. The episode also tells us the Enterprise is to be decommissioned.
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Picard season 2 showed a young Picard playing with a small (Eaglemoss) model of a refit NX Class Starship.
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Picard season 3 showed us a refit NX Class Starship, that according to multiple production sources, with accompanying artwork, was intended to be the NX-01.**
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Strange New Worlds somewhat implied that other NX Class Starship were mothballed, other than the NX-01.
*It should be noted that the entity of the 2161 segments of this episode are a Holodeck recreation.
**
To my knowledge, this is ALL we know canonically about the NX-01's later years.